Papers

We investigate Hawking-like particle production in analogue gravity systems with superluminal modified dispersion relations. For a broad class of even, convex, and polynomially bounded dispersion relations, we show that the relevant outgoing modes are governed by an effective horizon induced by dispersive propagation. Extending the near-horizon S-matrix method beyond the purely sonic regime, we compute the Bogoliubov coefficients and demonstrate that, in the low-energy and adiabatic limits, they agree with the tunneling result obtained from the approximant ray. In both cases, the emission spectrum is controlled by an effective surface gravity associated to the effective horizon, leading to controlled deviations from exact thermality. Our results establish an analytical connection between the Bogoliubov and tunneling descriptions in dispersive settings and clarify the conditions under which Hawking radiation remains robust against ultraviolet modifications, with implications extending beyond analogue gravity.

We construct a Born-Infeld-type $f(R,{\cal G})$ modification of gravity, where ${\cal G}$ is the Gauss-Bonnet term, by embedding Born-Infeld electrodynamics in a five-dimensional pure modified gravity. This method leads to the correspondence between curvature scalars and electromagnetic field strength scalars – $R\leftrightarrow F_{\mu\nu}F^{\mu\nu}$ and ${\cal G}\leftrightarrow (\epsilon_{\mu\nu\rho\sigma}F^{\mu\nu}F^{\rho\sigma})^2$ – allowing us to replicate the structure of Born-Infeld electrodynamics in the gravitational sector. The resulting Born-Infeld-type gravity is a ghost-free $f(R,{\cal G})$ theory which reduces to Einstein gravity in the low energy limit. In this work we focus on bouncing cosmological solutions of such a theory, which require positive spatial curvature. By using both the Jordan and Einstein frame analyses, we find a vast space of bouncing solutions with different asymptotic behaviors, including solutions with multiple bounces grouped together. Observational consequences of such solutions will be investigated in the future.

We classify mobile Pauli stabilizer codes up to gapped interfaces and coarse-graining using the framework of algebraic $\mathrm{L}$-theory. We compare this classification with that of framed TQFTs, theories that arise naturally in the continuum, highlighting a close structural relationship between the two. Our approach is formulated in the category of perfect chain complexes equipped with quadratic functor over the Laurent polynomial ring $R = \mathbb{Z}/p[x_1^{\pm 1}, \ldots, x_n^{\pm 1}]$, within which the collection of topological operators of Pauli stabilizer codes arise naturally as objects. In particular, we establish a bulk-boundary correspondence for lattice theories: the equivalence class of a Pauli stabilizer code up to gapped interface is described by a Clifford QCA in one dimension higher. This is done using the universal target category for stabilizer codes, which is the categorical spectrum whose existence and universal properties are introduced in this work. We conclude by highlighting subtle differences between the classification of Pauli stabilizer codes and TQFTs, leading to qualitative distinctions between lattice and continuum theories.

We study three-gluon scattering amplitudes in global de Sitter spacetime, in the angular momentum basis of SO(1,4) symmetry representations. At the tree level, they are determined by the intertwiner integrals of harmonic one-forms on the three-sphere. We derive a general formula valid for all helicity configurations of incoming and outgoing gluons and express the amplitudes in terms of Wigner 3j symbols.

Motivated by the Swampland Distance Conjecture, we study distances in field space using the framework of Optimal Transport. The associated optimisation problem naturally leads to a notion of distance in terms of a (generalised) Wasserstein distance between probability distributions over field space. In the absence of dynamical gravity, we relate the transport problem to Hamilton-Jacobi and continuity equations arising from a WKB expansion of a Schrödinger equation associated with the physical configuration. We then formulate an extension in the presence of dynamical gravity. Using the ADM formalism, we establish the corresponding transport problem through the Wheeler-DeWitt equation, giving rise to different possible choices of cost functions. The resulting notions of distances are naturally defined on the full configuration space, while an interpretation in terms of a genuine scalar field distance requires additional modifications. We further discuss several applications and examples, and indicate possible implications for different themes within the Swampland program.

The three-dimensional cubic conformal field theory governs the critical behaviour of Heisenberg magnets with cubic anisotropy. Studying this theory non-perturbatively is challenging, because its most easily accessible observables are numerically very close to those of the more symmetric $O(3)$ model. In this work, we overcome this difficulty using the fuzzy sphere regularisation method. By adding a cubic-invariant two-body interaction to the quantum rotor Hamiltonian used for the $O(3)$ model, we break the continuous rotational symmetry by construction and unambiguously isolate the cubic critical point. Using exact diagonalisation and the density matrix renormalisation group, we calculate the scaling dimensions of several key operators, including the leading scalar singlets, and resolve the splitting of the $O(3)$ rank-two traceless symmetric tensor into the $E_g$ and $T_{2g}$ representations of the cubic group. Our results are consistent with existing Monte Carlo, conformal perturbation theory, and $\varepsilon$ expansion benchmarks, demonstrating the power of the fuzzy sphere in resolving closely spaced universality classes.

The method of extremal flows has presented an alluring alternative approach to numerically solving bootstrap constraints. Here I present the development and adaptation of that approach to a more general class of flows with apparent discontinuities. I focus on upgrading solutions of gap maximization for the spinning modular bootstrap from low to high numerical order, though the methodology is generic to a broader class of bootstrap constraints and flows. This methodology presents various nontrivialities and nuances which reflect a richness of the space of bootstrap solutions. The result is a prototype which successfully upgrades solutions in a simple test case at small scale.

Despite the archetypal status of the BTZ background in quantifying quantum aspects of black holes, several features at low temperatures remain imprecise and incomplete. Here, we systematically investigate the behaviour of the Euclidean path integral at low temperatures in the context of AdS3 supergravity, including an analysis of quantum fluctuations in both the near-horizon and asymptotic regions. We clarify and rectify aspects of the bosonic fluctuations, highlighting the role of boundary conditions in AdS3, and show in particular that the gravitational path integral in the near-horizon region is inequivalent to that around BTZ at low temperature. We further account in detail for the contributions of Chern-Simons fields and spin-3/2 modes, thereby refining the disparities between the near-extremal and near-BPS limits at low temperature. Altogether, our analysis sharpens the distinction between near- and far-region dynamics and demonstrates a disagreement in the gravitational path integral at the quantum level.

This work investigates the role of the $U(N) \times U(\tilde{N})$ global symmetry in tree-level scattering amplitudes of the bi-adjoint $\phi^3$ theory from three perspectives: combinatorics, correlation functions, and a massive extension of the CHY formalism. We derive a planar scattering potential whose extrema reproduce Dolan and Goddard's massive scattering equations, providing physical intuition of the construction. This potential enables the counting of kinematic invariants via maximally symmetric Ferrers shapes, and it is expressed in terms of conformally invariant cross-ratios. We find that the $U(1)$ decoupling identity provides a physical interpretation of two different Catalan recursion relations, and also reveals an interplay between Catalan and Narayana numbers in the $U(1)$ splitting. Finally, we construct correlation functions for a fixed particle ordering using the CHY formalism, offering new insights into the dynamics of such amplitude structures. We derive a closed form expression of the reduced number of solutions for this set-up, as well as an off-shell scattering potential.

We investigate whether retarded scalar propagators on causal sets can be expressed in terms of the link matrix $\mathbf{L}$. For Poisson sprinklings into $1+1$ dimensional Minkowski spacetime, we show by asymptotic analysis and supporting numerical simulations that the averaged massless retarded propagator is naturally associated with a normalized exponential exp$(\mathbf{L})$. We then extend the construction to the massive case via the usual mass-scattering series and obtain good agreement with the continuum propagator after averaging. Finally, we discuss the inverse kernel exp$(-\mathbf{L})$ as a possible candidate for a discrete d'Alembertian.

Conformal symmetry is expected to be realized in many equilibrium statistical mechanical systems at criticality. Although this is certainly true in two-dimensional systems, the three-dimensional case is subtler, and only a few proofs exist, only so in very specific cases. In this work, we give two proofs for the large $N$ limit of the $O(N)$ universality class within the non-perturbative renormalization group framework: one functional, and one vertex-by-vertex in Fourier space. While doing so, we unveil how the theory is structured in order for conformal symmetry to be realized. As a consequence, we shed light on what to expect, on rather general grounds, for a theory to be conformally invariant.

In this paper, we investigate the thermodynamic and radiative properties of a regular black hole sourced by a dark matter halo described by the Einasto density profile. The closed-form expressions for the Hawking temperature, specific heat capacity, sparsity parameter of Hawking flux, and the spectral energy emission rate were obtained. All these are examined as a function of the characteristic scale parameter $\alpha$ of the dark matter distribution and compared with the standard Schwarzschild results. We show that the presence of a dark matter halo suppresses both the Hawking temperature and energy emission rate relative to the standard black hole result. Crucially, the specific heat capacity can be positive for a finite range of horizon radii, signaling a thermodynamically stable phase; the boundary of this stable region defines a Davies-type phase transition. The sparsity parameter is higher than in the standard black hole, indicating that the dark matter environment makes the Hawking flux even more intermittent.

Numerical analysis to determine the form of the scalar power spectrum in Warm Inflationary paradigm is inevitable. One further needs numerical techniques to analyse any Warm Inflation model with the current observational data through the MCMC codes that are available publicly, like COSMOMC or Cobaya. We present SWIM (Stochastic Warm Inflation Module) written in C++ and Python, that not only helps generate the Warm Inflationary scalar power spectrum, either semi-analytically or fully numerically, but also is integrated with Cobaya enabling the user to constrain the model parameters with current CMB data and thus to put any Warm Inflation model to test. SWIM numerically solves the standard stochastic perturbation equations of Warm Inflation without any approximations, uses machine learning techniques to speed up the MCMC analysis while analysing the fully numerical power spectrum that significantly reduces the computational cost, and is able to accommodate any Warm Inflation model with any form of inflationary potential and dissipative coefficient for numerical analysis. We show that SWIM, in most of the cases, outperforms other numerical codes on Warm Inflation that are designed to yield only the semi-analytical power spectrum as far as the runtimes are concerned. We further point out that there can be situations where the semi-analytical way of determining the scalar power spectrum in Warm Inflation can fall short, and one needs the full numerical power spectrum for parameter estimation given the observational data. In such cases, SWIM is the only code available so far that is designed to perform the task. Hence, SWIM offers a complete numerical platform for thorough analysis of Warm Inflation models against the current cosmological data. SWIM has been made publicly available at https://github.com/umg-kmr/SWIM.

In this work, we construct a three-dimensional non-relativistic Chern–Simons supergravity theory with both curvature and torsion within the Mielke–Baekler framework. We show that a consistent non-relativistic supergravity formulation requires starting from a $\mathcal{N}=2$ supersymmetric extension of the Mielke–Baekler algebra and implementing a non-relativistic expansion via the semigroup expansion method, rather than a naive contraction. This procedure allows one to overcome the usual difficulties of non-relativistic supergravity constructions, ensuring closure of the superalgebra and the existence of a non-degenerate invariant bilinear form. The resulting model is characterized by two parameters $(p,q)$, which interpolate between different non-relativistic supergravity theories, including the extended Bargmann, Newton–Hooke, and torsional models. Our results provide a unified framework for non-relativistic supergravity with torsion and open new avenues for exploring supersymmetric extensions of Galilean and Carrollian gravity theories.

We show that in critical loop models, torus 1-point functions can be expressed in terms of sphere 4-point functions at a different central charge. Unlike in the Moore–Seiberg formalism, crossing symmetry on the sphere therefore implies modular covariance on the torus. We systematically compute torus 1-point functions in critical loop models, using a numerical bootstrap approach. We focus on the 1-point functions of the 6 simplest primary fields, which give rise to 10 solutions of modular covariance equations. Such 1-point functions are infinite linear combinations of conformal blocks. The coefficients are products of double Gamma functions, times polynomial functions of loop weights. For each solution, we determine the first 6 to 12 polynomials.

Through analyzing the quark propagator under the magnetic field, we establish that the axial anomaly originates from an additional Dirac structure in quark propagator induced by the magnetic field. This Dirac structure also allows one to connect the axial anomaly with the topological properties of the system by checking the axial vector Ward identity. For the tree level propagator, we reproduce the result of the anomalous axial current as in the Dirac Hamiltonian approach and kinetic theory. Particularly, we confirm that the chiral magnetic effect (CME) comes from the same term that is in charge of the axial anomaly, specifically, as the anomaly of the transversal axial vector Ward Identity. The identity guarantees that the CME conductivity $C_{\rm CME}$ is a constant as $C_{\rm CME}=\frac{1}{2\pi^2}$, and is robust against the temperature, chemical potential, magnetic field and also interaction. Finally, we verify this numerically by applying the full quark propagator under magnetic field calculated from the functional QCD methods.

We introduce a new regularization scheme for divergent integrals in quantum field theory. The framework is based on the structural decomposition of the integrand asymptotic expansion, which distinguishes between contributions that drive UV singularities and those that remain finite. This asymptotic regularization method isolates the genuinely singular sector and enables a consistent subtraction of divergences while maintaining covariance and gauge symmetry. In single-scale theories, we show that the renormalized quantities exhibit a non-local logarithmic dependence uniquely determined by the UV asymptotics, offering a derivation of logarithmic terms that is independent of standard renormalization-group flows. Because it relies only on asymptotic structure rather than on standard relativistic power counting, the method is naturally applicable to theories with modified dispersion relations and non-standard UV scaling. Although formulated here for ultraviolet divergences, the underlying strategy extends straightforwardly to infrared singularities.

We study a class of nonlinear effective field theories (EFTs) in which relative entropy organizes an infinite tower of higher-dimensional operators with factorially growing coefficients. Using its Borel-resummed expression, we derive bounds on EFT coefficients: under the factorial-growth ansatz, the non-negativity of relative entropy in the weak-background regime constrains the sign of their assumed asymptotic growth, while a negative real part beyond this regime signals perturbative limitations and possible sensitivity to nonperturbative effects. In fermionic QED, analytic continuation from Euclidean to Minkowski spacetime yields a controlled one-loop example: such negativity accompanies the Schwinger instability.

Effective field theory (EFT) provides a systematic framework to describe deviations from general relativity through higher-curvature corrections to the gravitational action. In this work, we investigate test-field quasinormal modes (QNMs) and weak and strong gravitational lensing in static, spherically symmetric spacetimes with higher-curvature corrections, focusing on null geodesics near the photon sphere. Using the strong deflection limit formalism developed by Bozza, we derive the logarithmic divergence of the deflection angle and separate the divergent and regular contributions. We normalize the dimensionful EFT couplings by appropriate powers of the horizon radius $r_s=2M$ and work perturbatively in the resulting dimensionless parameters. We analyze how higher-curvature corrections modify the photon sphere radius, the critical impact parameter, and the coefficients governing the strong deflection expansion. We further examine the corresponding effects on test-field QNMs in the eikonal regime. Our results provide a systematic characterization of how higher-curvature EFT corrections modify null geodesics, strong gravitational lensing, and test-field QNMs, and clarify the relation between EFT couplings and characteristic properties of the surrounding spacetime geometry.

Landau models serve as quantum mechanical systems for generating quantum matrix geometries. In this paper, we demonstrate that Howe duality provides the underlying structure of the super Landau model, reflecting a general feature of coset-type Landau models. The (super) Howe duality relates different Landau levels and accounts for the emergence of a dual fuzzy geometry. By employing super-spinor derivative operators, the supermonopole harmonics in both integer and half-integer Landau levels are explicitly constructed and the algebraic structure of the super-Hilbert space is revealed. We propose a consistent probabilistic interpretation for these wavefunctions defined on a supermanifold. Through a level projection method, we derive the matrix coordinates of fuzzy supersphere geometries for arbitrary Landau levels, along with a precise determination of the non-commutative scale factor. It is shown that the theta correspondence of Howe duality induces a geometric transformation between fuzzy objects. Finally, we point out that Howe duality realizes an internal-external space duality and underlies quantum matrix geometries, suggesting that it may play a fundamental role in understanding Matrix model geometries.

We present a unified framework for the discussion of black hole thermodynamics of $d$-dimensional static black holes with spherical or compact hyperbolic horizon topology satisfying $g_{tt}g_{rr}=-1$ in Schwarzschild gauge. To that end, we consider any such black hole as a solution to an integrable $2$-dimensional effective dilaton theory and thereby as a vacuum solution to a generalised notion of $d$-dimensional quasi-topological gravity. We show that the generating function determining $f(r) = -g_{tt} $ in the integrated equation of motion provides the thermodynamic mass in a generalised first law with entropy computed as the Wald entropy. The framework presented here can be applied to singular and regular black holes with flat or anti-de Sitter asymptotics.

Using the gauge-gravity duality, we construct the anisotropic D3/D7 approach as a three-dimensional QCD-like theory, then investigate systematically the hadronic mass spectra, the dragging terms and the lowest hadronic interactions in the presence of the anisotropy in holography. Our derivation illustrates the dragging terms in the effective action are very necessary for an anisotropic theory since they are the key roles to affect the transport properties of the dual theory. And the numerical results in addition imply the hadronic system will become unstable if the anisotropy is much larger than its confinement energy scale. It agrees with that the confining phase in this approach becomes unstable if the anisotropy is sufficiently large in our previous works with this model. Therefore, this work is constructive to understand the anisotropy in the gauge field theory.

A renormalization-group (RG) improved Schwarzschild-like black hole (BH) is studied here, with a lapse that interpolates between a classical Schwarzschild exterior and a quantum-smoothed interior set by a cutoff scale $\xi$ and an interpolation parameter $\gamma$. We work out the horizon structure together with the photon sphere and shadow radius $R_{\mathrm{sh}}$, set up the scalar, electromagnetic, and Dirac Regge-Wheeler-Zerilli problems in a single treatment, and compute the fundamental and overtone quasinormal modes by sixth-order WKB, cross-checked against time-domain ringdown. For $\xi>0$ and $\gamma>0$ the geometry is regular, with a de Sitter core. Strong Cosmic Censorship (SCC) is examined at the inner Cauchy horizon, which the improved geometry generates without charge or rotation. The quasinormal spectral gap $\beta=|\mathrm{Im}\,\omega|/\kappa_-$ stays multipole-independent at the $6\%$ level and follows $\beta\simeq\lambda_{L}/(2\kappa_{-})$. It remains below the de Sitter Christodoulou bound across the parameter range, and the asymptotically flat late-time tail places the geometry in the SCC-respecting class. A thermodynamic analysis identifies a Davies-type phase transition of the outer horizon, with the Schwarzschild $T_{H}\propto 1/r_{+}$ decay replaced by a bell-curve profile peaking at $T_{H}^{\max}\simeq 0.062$. A scan of the $(\xi,\gamma)$ plane gathers the joint behavior of the shadow, the scalar barrier, the SCC ratio, and $T_H$. Set against Bardeen, Hayward, and Bonanno-Reuter BHs at matched perturbation scale, the improved Schwarzschild BH is the most Schwarzschild-like of the regular-BH family, its static shadow radius degenerate with Hayward and Bonanno-Reuter at the percent level. The closing analysis takes up the sparsity of the Hawking flux and the energy-emission rate, both tied to the outer-horizon surface gravity through a single auxiliary function.

We show that the Weyl (trace) anomaly gives rise to a new non-dissipative vector current in accelerated relativistic fluids. The anomaly uniquely fixes the second-order transport coefficient governing the coupling between the electromagnetic field and the fluid acceleration. We derive this result by extending hydrodynamic anomaly matching to include the trace anomaly, and independently reproduce it in boundary quantum field theory by treating the Rindler horizon of an accelerated observer as an effective boundary. From the boundary perspective, the electric- and magnetic-field sectors correspond to screening and vacuum magnetization effects near the boundary. In the local rest frame, the electric-field contribution induces an additional charge density, while the magnetic-field contribution generates a transverse current with a Nernst-like, more generally thermomagnetic Hall-like, tensor structure. Our results reveal a new class of anomaly-induced transport governed by the trace anomaly.

We study the thermodynamics and interior structures of dynamical end of the world (EoW) branes in the rotating BTZ black hole. By mapping the induced metric of the branes to an effective Jackiw-Teitelboim (JT) system, we derive the first law of thermodynamics for the boundary conformal field theory (BCFT), incorporating boundary degrees of freedom. To construct this thermodynamics, we adopt two frameworks of black hole chemistry and the duality between the JT black hole and the Sachdev-Ye-Kitaev (SYK) model. In addition, we verify that the shadow entropy is equivalent to the boundary entropy via Hubeny-Ryu-Takayanagi (HRT) surfaces. Furthermore, we explore the possible interior configurations of the EoW brane inside the horizon. Two representative configurations, namely single and double-joint EoW branes, are considered. These disparate configurations share the same exterior brane configuration outside the horizon. By comparing their energies, we show that a transition could occur within the horizon.

We review recent developments on the QCD phase diagram at small chemical potentials and increasing temperature. There are three regimes/phases in QCD which differ by symmetries, degrees of freedom and N_c scaling: the hadron gas below the chiral restoration temperature T_ch, the stringy fluid between T_ch and the deconfinement temperature T_d and the quark-gluon plasma above T_d.

In this paper, we propose a universal diagrammatic interpretation of hidden zeros and $2$-splits of tree-level amplitudes. Originally developed for ${\rm Tr}(\phi^3)$ amplitudes in our previous work, this interpretation is now extended to tree-level amplitudes in Nonlinear sigma model (NLSM) and Yang-Mills (YM) theories. The interpretation is based on a certain factorization behavior of Feynman diagrams under specific kinematic constraints, which we term shuffle factorization along a specific line (SFASL). This mechanism allows us to separate Feynman diagrams along specific lines after summing over shuffle permutations. When applied to NLSM and YM amplitudes, we perform proper extensions of the SFASL used in the ${\rm Tr}(\phi^3)$ case. Through the SFASL, the interpretation for the hidden zeros and $2$-splits of tree amplitudes of ${\rm Tr}(\phi^3)$, NLSM, and YM can be unified as: the hidden zeros are ascribed to the on-shell condition $k_j^2=0$ of a massless particle, while the $2$-splits are caused by separating each Feynman diagram along two lines, akin to unzipping two zippers.

We study the small-$c$ expansion of general relativity in ADM variables up to next-to-next-to-leading order (NNLO). We show that, in the stationary sector, this formulation renders the field equations more tractable for explicit solution building. The stationary sector exhibits both strong-gravity and weak-field branches, whose structure becomes richer at NNLO. In the strong-gravity branch, we first obtain exact vacuum solutions of NLO Carroll gravity, including the Lense–Thirring and rotating C-metric backgrounds. At NNLO, we then construct the corresponding Lense–Thirring-type and C-metric-type exact vacuum geometries. These solutions also arise from the small-$c$ expansion of the Kerr and rotating C-metric geometries around the strong-gravity background, up to $\mathcal{O}(J)$ at NLO and up to $\mathcal{O}(J^3)$ at NNLO. In the weak-field branch, we find exact Hartle–Thorne-type solutions with an independent quadrupole moment, together with exact spin-squared corrections and a mixed quadrupolar-rotating solution. We further extend the $\ell=0,2$ sector by including higher multipoles up to $\ell=4$, where $\ell$ denotes the multipole index. These results show that the full NLO/NNLO theory admits a richer stationary vacuum sector than the magnetic Carroll truncation. More broadly, the ADM formulation provides a practical framework for constructing and analyzing stationary backgrounds in the small-$c$ expansion of general relativity, and may also offer a useful framework for organizing rotational and higher-multipole deformations motivated by relativistic compact-object geometries.

Let $A:=\mathbb{C}[z_+,z_-]\otimes \Lambda(\theta_1,\theta_2,\theta_3)$, with $z_\pm$ even and $\theta_1,\theta_2,\theta_3$ odd. For a reductive Lie algebra $\mathfrak g$, let $\mathfrak g[A]:=\mathfrak g\otimes A$ be the corresponding current Lie superalgebra. Motivated by the Chang–Yin description of weak-coupling $1/16$-BPS cohomology in $\mathcal N=4$ super-Yang–Mills, we study the relative Lie algebra cohomology $H^\bullet(\mathfrak g[A],\mathfrak g;\mathbb{C})$. We isolate three finite-rank phenomena. First, the natural $3|2$ super-commuting restriction map, viewed as a super analogue of Chevalley restriction and its commuting-scheme variants, already fails to be an isomorphism for $\mathfrak g=\mathfrak{so}_7$; the obstruction is a non-Cartan class. Second, the same algebra produces explicit fortuitous classes for $\mathfrak{sl}_2$ and $\mathfrak{so}_7$, giving concrete counterexamples to naive stable-image expectations suggested by the type-A Loday–Quillen–Tsygan theorem and its current-algebra refinements. Third, the classical relative cohomologies for the Langlands-dual pair $(\mathfrak{so}_7,\mathfrak{sp}_6)$ are not isomorphic. We then record the conjectural quantum deformation of the differential expected to restore duality, together with first-order evidence pairing the fortuitous and non-Cartan $\mathfrak{so}_7$ classes.

We reconsider the problem of regularizing the divergent series $\sum_{n=1}^{\infty}n^{\alpha}$ for $\operatorname{Re}\alpha>-1$, and offer a regularization prescription that yields the Riemann zeta regularization as a special case. The development of the regularization is framed as a two-step problem. The first step is prescribing a regularization of the divergent sum $\sum_{n=1}^{\infty}n^m$ for every non-negative integer $m$; and the second step is the extension of the sum for non-integer $\alpha$. The extension is obtained under the consistency condition that the regularized sum for integer $m$ emerges continuously from the sum for non-integer $\alpha$. The scheme is specified by a differential generator $L=L(\mathrm{d}/\mathrm{d}t)$ through which a generalized spectral function (GSF), $K_L(t)$, is constructed. Under the condition that the GSF has a holomorphic complex extension $K_L(z)$ with $z=0$ as a pole, the case for integer $m$ takes the regularized value $\sum_{n=1}^{\infty} n^m = (2\pi i)^{-1}\oint_C L^m K_L(z) z^{-1}\mathrm{d}z$, where $C$ is a closed contour enclosing only the pole of $K_L(z)$ at the origin. On the other hand, under the consistency condition, the case for non-integer $\alpha$ takes the value $\sum_{n=1}^{\infty}n^{\alpha}=(2\pi i)^{-1}\int_{\tilde{C}} L^{\alpha} K_L(z) z^{-1}\mathrm{d}z$, where $L^{\alpha}$ is the fractional extension of $L^m$ and $\tilde{C}$ is an appropriate deformation of the contour $C$. Here, we obtain the regularization corresponding to the generator $L=h(t) \mathrm{d}/\mathrm{d}t$, with $h(t)$ positive for all $t>0$, monotonically non-increasing, and admitting complex extension $h(z)$ such that $1/h(z)$ is entire. We find that the regularized sum is equal to the Riemann zeta regularized value plus terms determined by the generator $L$.